Trace and Künneth formulas for singularity categories and applications

We present an $\ell$ -adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas dg-category of singularities inertia-invariant vanishing cycles. As application, categorical version Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that monodromy action inertia group is unipotent.